Trigonometry Examples

$\log (4 x^{2} + 5 x) = \log (34 x + 63)$

Step 1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$4 x^{2} + 5 x = 34 x + 63$

Step 2

Solve for $x$ .

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Step 2.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.1.1

Subtract $34 x$ from both sides of the equation.

$4 x^{2} + 5 x - 34 x = 63$

Step 2.1.2

Subtract $34 x$ from $5 x$ .

$4 x^{2} - 29 x = 63$

Step 2.2

Subtract $63$ from both sides of the equation.

$4 x^{2} - 29 x - 63 = 0$

Step 2.3

Factor by grouping.

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Step 2.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 63 = - 252$ and whose sum is $b = - 29$ .

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Step 2.3.1.1

Factor $- 29$ out of $- 29 x$ .

$4 x^{2} - 29 x - 63 = 0$

Step 2.3.1.2

Rewrite $- 29$ as $7$ plus $- 36$

$4 x^{2} + (7 - 36) x - 63 = 0$

Step 2.3.1.3

Apply the distributive property.

$4 x^{2} + 7 x - 36 x - 63 = 0$

Step 2.3.2

Factor out the greatest common factor from each group.

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Step 2.3.2.1

Group the first two terms and the last two terms.

$(4 x^{2} + 7 x) - 36 x - 63 = 0$

Step 2.3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x + 7) - 9 (4 x + 7) = 0$

Step 2.3.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 7$ .

$(4 x + 7) (x - 9) = 0$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x + 7 = 0$

$x - 9 = 0$

Step 2.5

Set $4 x + 7$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $4 x + 7$ equal to $0$ .

$4 x + 7 = 0$

Step 2.5.2

Solve $4 x + 7 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $7$ from both sides of the equation.

$4 x = - 7$

Step 2.5.2.2

Divide each term in $4 x = - 7$ by $4$ and simplify.

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Step 2.5.2.2.1

Divide each term in $4 x = - 7$ by $4$ .

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{4}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{4}$

Step 2.6

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.6.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.7

The final solution is all the values that make $(4 x + 7) (x - 9) = 0$ true.

$x = - \frac{7}{4}, 9$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{4}, 9$

Decimal Form:

$x = - 1.75, 9$

Mixed Number Form:

$x = - 1 \frac{3}{4}, 9$